Kara Sears
The concept of flux and laws are intertwined in the field of physics, with flux describing the magnitude and direction of the flow of a substance or energy through a surface, and laws providing the mathematical framework to understand and calculate this flow. In the case of Gauss's law, also known as Gauss's flux theorem, it was formulated by Carl Friedrich Gauss in 1835 and published in 1867, providing a relationship between the distribution of electric charge and the resulting electric field. Fick's laws of diffusion, on the other hand, were first introduced by Adolf Fick in 1855 and describe the diffusion process of particles from high to low concentration. Flux, as a concept, has been integral to the development of various laws and theories, including those by Joseph Fourier, Isaac Newton, and Albert Einstein, who unified electric and magnetic forces under the electromagnetic force.
| Characteristics | Values |
|---|---|
| Flux | A concept in applied mathematics and vector calculus with many applications in physics |
| The rate of volume flow across a unit area (m3·m−2·s−1) | |
| The rate of movement of molecules across a unit area (mol·m−2·s−1) | |
| The rate of heat flow across a unit area (J·m−2·s−1) | |
| Laws | Rules that govern behaviour |
| Fick's laws of diffusion describe diffusion and were first posited by Adolf Fick in 1855 | |
| Gauss's law, also known as Gauss's flux theorem, is a law relating the distribution of electric charge to the resulting electric field |
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What You'll Learn
Fick's laws of diffusion
Fick's first law of diffusion describes the movement of particles from high to low concentration (diffusive flux) and how it is directly proportional to the particle's concentration gradient. The law can be used to derive Fick's second law, which is identical to the diffusion equation. The first law can be applied to systems in which the conditions remain the same, i.e., the flux coming into the system equals the flux going out. It can be used to predict the flux of reactants to the substrate and product away from the substrate. It also has applications in radiation transfer equations.
Fick's second law predicts the change in concentration gradient with time due to diffusion. This law is more applicable to physical science and other systems that are changing, i.e., non-steady-state systems. It is applied when the solution is not equal throughout.
Fick's laws can be used to solve for the diffusion coefficient, D. The diffusion coefficient can be solved for using Fick's first law, which relates the diffusive flux to the gradient of the concentration. The diffusion coefficient has units of m2 s, and its dimension is area per unit time. The driving force for one-dimensional diffusion is the quantity −∂φ/∂x, which for ideal mixtures is the concentration gradient.
Fick's laws have applications in various fields, including the fabrication of semiconductors, food industries, and chemical systems.
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Gauss's law
Electric Flux and Closed Surfaces
Electric flux, a measure of the strength of an electric field passing through a surface, is defined as the integral of the electric field. Gauss's law states that the electric flux through any closed surface is equal to the total charge enclosed within that surface divided by the permittivity of free space (also known as the vacuum or vacuum permittivity), represented by ε₀. Mathematically, this relationship can be expressed as:
> \ \[ \Phi = \oint_S \vec{E} \cdot \hat{n} dA = \frac{q_{enc}}{\epsilon_0} \]
Here, Φ represents the electric flux, \\(\vec{E}\) is the total electric field, and \(q_{enc}\) is the total charge enclosed by the closed surface S.
This law applies regardless of the shape or size of the closed surface, as long as it encloses the same charge. If there is no net charge within the closed surface, then the electric flux passing through it is zero.
Applications and Extensions
Example: Spherical Charge Distribution
When dealing with a spherical charge distribution, Gauss's law becomes particularly useful. By choosing a Gaussian surface with the same symmetry as the charge distribution, one can determine the electric field at a specific point outside or inside the distribution. This is achieved by evaluating the integral of the electric field over the Gaussian surface.
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Faraday's law of induction
Michael Faraday, an English scientist, proposed the law of electromagnetic induction through a series of experiments in 1831. In one of his experiments, Faraday wrapped two coils of wire around an iron ring and connected one coil to a battery. He observed a brief deflection in a galvanometer attached to the second coil and concluded that a changing current in the first coil created a changing magnetic field in the ring, which induced a current in the second coil.
Faraday's first law of electromagnetic induction states that when a conductor is placed in a varying magnetic field, an electromotive force (emf) is induced. If the conductor circuit is closed, a current is induced, known as an induced current. The second law quantifies the emf produced in the conductor and states that the induced emf in a coil is equal to the rate of change of flux linkage.
The direction of the induced emf can be determined using Lenz's law, formulated by Emil Lenz in 1834, which describes the "flux through the circuit" and the resulting direction of the induced emf and current. Faraday's law is a crucial concept in understanding electromagnetic induction and has practical applications in various electrical devices, such as transformers, induction cookers, and musical instruments like electric guitars and violins.
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Fourier's law of conduction
The concept of heat flux was introduced by Joseph Fourier in his analysis of heat transfer phenomena. Fourier's law of conduction, also known as the law of heat conduction, describes the rate of heat transfer through a material.
According to Fourier's law, the rate of heat transfer is directly proportional to the negative gradient in temperature and the area at right angles to that gradient through which the heat flows. In other words, the law states that the conduction of heat occurs from a region of higher temperature to a region of lower temperature, with the rate of heat transfer depending on the temperature difference and the cross-sectional area for the heat to flow through.
The law can be expressed mathematically as:
> \(\begin{array}{l}Rate\,of\,heat\,conduction\propto \frac{(area)(temperature\;difference)}{thickness}\end{array}\)
Where:
- T1 and T2 represent the temperature difference across a small distance Δx
- A is the area through which the heat flows
- K is the thermal conductivity of the material
Fourier's law assumes two fundamental quantities: temperature and heat flow. It is a contextual law, implying that it is selective in its operation depending on the specific context and elements involved. For example, when considering the temperature of a tissue in an organism, the temperature of the constituent organs and organelles cannot be different from that of the tissue itself.
Fourier's law is a fundamental principle in understanding and calculating conduction heat transfer, and it has found applications in various fields, including the development of functionally graded materials and thermal metamaterials.
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Maxwell's equations
In the context of the search results, "flux" and "laws" refer to concepts in physics.
Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. It is a concept in applied mathematics and vector calculus with many applications in physics. In physics, flux is a vector quantity, describing the magnitude and direction of the flow of a substance or property.
Now, onto Maxwell's equations. These are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, optics, and electric and magnetic circuits. The equations are named after the physicist and mathematician James Clerk Maxwell, who, in 1861 and 1862, published an early form of the equations that included the Lorentz force law.
The equations define two new auxiliary fields that describe the large-scale behaviour of matter without considering atomic-scale charges and quantum phenomena. The term "Maxwell's equations" is also used for equivalent alternative formulations, and these equations are useful for solving boundary value problems, analytical mechanics, and quantum mechanics.
The modern form of the equations in their most common formulation is credited to Oliver Heaviside, whose vector calculus formalism has become the standard. This formalism is rotationally invariant and, therefore, mathematically more transparent than Maxwell's original 20 equations in x, y, and z components.
One of Maxwell's equations is Gauss's law, which describes the relationship between an electric field and electric charges. Gauss's law, or Gauss's flux theorem, states that the flux of the electric field out of a closed surface is proportional to the electric charge enclosed by the surface, irrespective of how that charge is distributed.
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Frequently asked questions
Flux describes any effect that appears to pass or travel through a surface or substance. It is a concept in applied mathematics and vector calculus with many applications in physics.
There are several laws associated with flux, including Gauss's law, Fick's laws of diffusion, Fourier's law of conduction, Darcy's law of groundwater flow, and Faraday's law of induction.
Flux came first. The concept of heat flux was introduced by Joseph Fourier in his analysis of heat transfer phenomena. The term “fluxion” was first introduced into differential calculus by Isaac Newton.
Gauss's law, also known as Gauss's flux theorem, relates the distribution of electric charge to the resulting electric field. It states that the flux of the electric field out of a closed surface is proportional to the electric charge enclosed by the surface.
Fick's laws of diffusion describe diffusion and were first posited by Adolf Fick in 1855 based on experimental results. Fick's first law predicts the flux of reactants and products, while Fick's second law predicts changes in the concentration gradient over time due to diffusion.
